This work is concerned with the reconstruction of the original undistorted image of a curled document or book when captured face-up by a camera with the support of sparse depth measurements. A novel method based on physically modeling paper deformation with an applicable surface is proposed and a relaxation algorithm is described that allows us to fit it to noisy data and then flatten it in order to produce the final undistorted image. The promising results obtained confirm that the use of applicable surfaces is the right way to address this problem.